Equivariant filter (EqF): A general filter design for systems on homogeneous spaces

TL;DR

This paper introduces the Equivariant Filter (EqF), a novel filter design leveraging symmetry properties of systems on homogeneous spaces, demonstrated on landmark position estimation with a monocular camera.

Contribution

The paper presents a new filter design, the EqF, that exploits system equivariance on homogeneous spaces, extending previous symmetry-based filtering approaches.

Findings

Successfully applied to landmark position estimation problem.

Outperforms previous symmetry-based filters in intractable scenarios.

Demonstrates the utility of equivariance in filter design.

Abstract

The kinematics of many mechanical systems encountered in robotics and other fields, such as single-bearing attitude estimation and SLAM, are naturally posed on homogeneous spaces: That is, their state lies in a smooth manifold equipped with a transitive Lie-group symmetry. This paper shows that any system posed in a homogeneous space can be extended to a larger system that is equivariant under a symmetry action. The equivariant structure of the system is exploited to propose a novel new filter, the Equivariant Filter (EqF), based on linearisation of global error dynamics derived from the symmetry action. The EqF is applied to an example of estimating the positions of stationary landmarks relative to a moving monocular camera that is intractable for previously proposed symmetry based filter design methodologies.